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In my calculation (all the possible numbers for these four digits) takes up to nine steps to reach 6174 ; which itself is a multiple of 9.
Now why only this takes up to ninth step may be important.

I don't know why; but I have a feeling that it has to do something with the following math puzzle:

1. Take up a number of any digit length.
2. Leave 18 rows empty.
3. write in the 20th row sum of all these unfilled numbers.
4. The sum would always be reached by adding 9 to left of the number in first row; taking rest of the digits same till tens then subtract 9 from the ones. ( Or You may require to adjust the tens with this subtraction.)
5. Then ask the person you're playing with to write another number in second row of same digit length as first.
6. Then in third row you write your number as to make the sum of two rows as 9,99,999,9999....

For Instance:
My friend take up a number 2986.
I would leave another 18 row empty. In row # 20 I write the answer: 92977
now in second row my friend wrote 3342; while I write in 6657 in row # 4 (sum of row # 3 & 4 is 9999)
then repeat the same process 09 more times (first he write a number then I write mine to make the two row sum to be 9999)
Now go check the total which was written long before the whole series of numbers has been written.

Please let me know; if this has something to do with 6174. And 6174 divided by 9 gives 686. may be this has something to do too.
Because each new derived number is multiple of nine and hence we reach multiple 686 times to get to 6174.